The levels of uncertainty, types of intervals

What are we uncertain about?

• Parameters prior to data (posterior)
• Parameters after data (posterior)
• Expectations (before or after data) given parameter uncertainty
• Observations given sampling variability only
• Future (potential) observations given parameter uncertainty and sampling variability

Some of the effort comes from complex models, interactions, and distributions

What does \(k\) mean or do in the equation: \(ks \ln\left(1+\frac{\beta I}{k}\right)\)?

What does an interaction between dose and strain mean in a logistic regression? How does this change if infection is likely or unlikely?

What does the shape parameter, \(\alpha\) do in a Gamma distribution?

Parameter uncertainty prior to data

This (un)certainty is reflected in prior distributions

What priors say about the distribution of (unobserved) data can be hard to grok

Parameter uncertainty prior to data

What priors say about the distribution of (unobserved) data can be hard to grok

\(\therefore\) use prior predictive simulation

Paramter uncertainty after data

This (un)certainty is reflected in the posterior distributions

To the extent parameters are meaningful/interpretable on their own (e.g., a slope) you can construct CIs and HDPIs, and other features of the marginal posterior distributions

What they say about expectations on the response scale can be hard to grok

Paramter uncertainty after data

What they say about expectations on the responses scale can be hard to grok

\(\therefore\) plot posterior inference (e.g., against data)

Uncertainty about future data

Involves parameter uncertainty + sampling noise.

What they say about the distribution of (future) data can be hard to grok

Uncertainty about future data

What they say about the distribution of (future) data can be hard to grok

\(\therefore\) use posterior predictive simulation